1. Fundamentals of Music Processing
Chapter 2: Fourier Analysis of Signals
Meinard Müller
International Audio Laboratories Erlangen

2. Book: Fundamentals of Music Processing
Meinard Müller
Fundamentals of Music Processing
Audio, Analysis, Algorithms, Applications
483 p., 249 illus., 30 illus. in color, hardcover
ISBN:
Springer, 2015
Accompanying website:

3. Chapter 2: Fourier Analysis of Signals
2.1 The Fourier Transform in a Nutshell
2.2 Signals and Signal Spaces
2.3 Fourier Transform
2.4 Discrete Fourier Transform (DFT)
2.5 Short-Time Fourier Transform (STFT)
2.6 Further Notes
Important technical terminology is covered in Chapter 2. In particular, we approach the Fourier transform—which is perhaps the most fundamental tool in signal processing—from various perspectives. For the reader who is more interested in the musical aspects of the book, Section 2.1 provides a summary of the most important facts on the Fourier transform. In particular, the notion of a spectrogram, which yields a time–frequency representation of an audio signal, is introduced. The remainder of the chapter treats the Fourier transform in greater mathematical depth and also includes the fast Fourier transform (FFT)—an algorithm of great beauty and high practical relevance.

4. 2 Fourier Analysis of Signals
Teaser

5. 2.1 The Fourier Transform in a Nutshell
Fig. 2.1

6. 2.1 The Fourier Transform in a Nutshell
Fig. 2.1
Time (seconds)
Time (seconds)

7. 2.1 The Fourier Transform in a Nutshell
Fig. 2.1
Time (seconds)
Time (seconds)

8. 2.1 The Fourier Transform in a Nutshell
Fig. 2.1
Time (seconds)
Time (seconds)

9. 2.1 The Fourier Transform in a Nutshell
Fig. 2.1
Time (seconds)
Time (seconds)

10. 2.1 The Fourier Transform in a Nutshell
Fig. 2.2
(a)
(b)
(b)
(c)
(a)
(c)
(d)
(d)

11. 2.1 The Fourier Transform in a Nutshell
Fig. 2.3

12. 2.1 The Fourier Transform in a Nutshell
Fig. 2.4

13. 2.1 The Fourier Transform in a Nutshell
Fig. 2.5

14. 2.1 The Fourier Transform in a Nutshell
Fig. 2.6

15. 2.1 The Fourier Transform in a Nutshell
Fig. 2.7

16. 2.1 The Fourier Transform in a Nutshell
Fig. 2.8

17. 2.1 The Fourier Transform in a Nutshell
Fig. 2.8
Time (seconds)
Frequency (Hz)

18. 2.1 The Fourier Transform in a Nutshell
Fig. 2.8
Time (seconds)
Frequency (Hz)

19. 2.1 The Fourier Transform in a Nutshell
Fig. 2.8
Time (seconds)
Frequency (Hz)

20. 2.1 The Fourier Transform in a Nutshell
Fig. 2.8
Time (seconds)
Frequency (Hz)

21. 2.1 The Fourier Transform in a Nutshell
Fig. 2.9

22. 2.1 The Fourier Transform in a Nutshell
Fig. 2.10

23. 2.1 The Fourier Transform in a Nutshell
Fig. 2.10
Frequency (Hz)
Time (seconds)

24. 2.1 The Fourier Transform in a Nutshell
Fig. 2.10
Frequency (Hz)
Time (seconds)

25. 2.2 Signals and Signal Spaces
Fig. 2.11

26. 2.2 Signals and Signal Spaces
Fig. 2.12

27. 2.2 Signals and Signal Spaces
Fig. 2.13

28. 2.2 Signals and Signal Spaces
Fig. 2.13
Amplitude
Time (seconds)

29. 2.2 Signals and Signal Spaces
Fig. 2.13
Amplitude
Sampling period
Time (seconds)

30. 2.2 Signals and Signal Spaces
Fig. 2.13
Amplitude
Sampling period
Time (seconds)

31. 2.2 Signals and Signal Spaces
Fig. 2.13
Amplitude
Quantization
step size
Sampling period
Time (seconds)

32. 2.2 Signals and Signal Spaces
Fig. 2.13
Amplitude
Quantization
step size
Sampling period
Time (seconds)

33. 2.2 Signals and Signal Spaces
Fig. 2.14

34. 2.2 Signals and Signal Spaces
Fig. 2.14
Amplitude
Time (seconds)

35. 2.2 Signals and Signal Spaces
Fig. 2.14
Amplitude
Time (seconds)

36. 2.2 Signals and Signal Spaces
Fig. 2.14
Amplitude
Time (seconds)

37. 2.2 Signals and Signal Spaces
Fig. 2.14
Amplitude
Time (seconds)

38. 2.2 Signals and Signal Spaces
Fig. 2.15

39. 2.3 Fourier Transform
Fig. 2.16

40. 2.3 Fourier Transform
Fig. 2.17

41. 2.3 Fourier Transform
Fig. 2.17

42. 2.3 Fourier Transform
Fig. 2.17

43. 2.3 Fourier Transform
Fig. 2.17

44. 2.3 Fourier Transform
Fig. 2.18

45. 2.3 Fourier Transform
Fig. 2.19

46. 2.3 Fourier Transform
Fig. 2.20

47. 2.3 Fourier Transform
Fig. 2.21

48. 2.3 Fourier Transform
Fig. 2.22

49. 2.3 Fourier Transform
Fig. 2.23

50. 2.3 Fourier Transform
Fig. 2.24

51. 2.3 Fourier Transform
Fig. 2.25

52. 2.3 Fourier Transform Fig. 2.25 Time (seconds) Frequency (Hz)

53. 2.3 Fourier Transform Fig. 2.25 Time (seconds) Frequency (Hz)

54. 2.4 Discrete Fourier Transform (DFT)
Table 2.1

55. 2.4 Discrete Fourier Transform (DFT)
Fig. 2.26

56. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.27

57. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.27
Time (seconds)
Frequency (Hz)

58. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.27
Time (seconds)
Frequency (Hz)

59. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.27
Time (seconds)
Frequency (Hz)

60. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.27
Time (seconds)
Frequency (Hz)

61. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.28

62. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.29

63. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.30

64. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.31

65. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.32

66. 2.5 Short-Time Fourier Transform (STFT)
Fig. 2.33

67. 2.6 Further Notes
Table 2.2